Usually when a 3-way switch fails, lights can be toggled on and off at one switch, but not the other.If one of the two 3-way switches toggles the light(s) on and off, the other 3-way switch has probably failed.
Turn off power to the switches at the circuit breaker and using a tester, make sure power is off. Remove the two 3-way switches so you can access the switch terminals. The wire connections do not have to be removed.
Using a continuity tester, check continuity between the common (black) terminal and each of the traveler terminals (brass colored) of each 3-way switch.
There should be continuity between the common terminal and only one of the traveler terminals. After toggling the switch, there should be continuity between the common terminal and only the other traveler terminal.
When a 3-way switch fails, there will usually be no continuity between the common terminal and one (sometimes both) traveler terminals of one of the 3-way switches.
How to Troubleshoot Three-Way Switches
Place leads of continuity tester on the common terminal and one of the traveler terminals
Place leads of continuity tester on the common terminal and the other traveler terminal